Despite the lack of chances, both sides appeared desperate to get onto the front foot throughout a tight first half in Milton Keynes.
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The Dons were close to taking the lead just before the break, but the home fans were left disappointed when Howard got a crucial hand to Tom Flanagan's header.
Karl Robinson's side reacted to the scare by taking control of the match, with the lively Afobe firing straight at Howard after beating a couple of United defenders.
Afobe took on the responsibility from 12 yards and he made no mistake as he handed the Dons the lead with a calm finish.
United responded well, but the home side appeared to be on course for victory after surviving an extended spell of pressure.
However, the Dons were unable to prevent a dramatic finish as Jose Baxter sprinted away from his marker before crossing to Higdon, who headed home in the 86th minute to keep the tie alive.
Extra time looked to be an inevitability, but there would be a late twist to the match in stoppage time as Higdon arrived at the back post to fire home from Jamie Murphy's cross and send his side through to round five.